Puree apple, shallot, and garlic with vinegar and salt in a blender, scraping down sides several times, until very smooth. With motor running, add oil in a slow stream, then blend until emulsified and very smooth, about 30 seconds. Gently stir together crab, mango, apple, cilantro, and 6 tablespoons dressing, then season with salt. Just before serving, put 1 apple chip on each of 6 plates and top each chip with about 3 tablespoons crab salad. Layer with another chip and another 3 tablespoons crab salad, then top each stack with 1 more chip. Drizzle plates with some of remaining dressing and serve immediately.